N5 Prelim Revision Booklet - Web Version

N5 Formula Sheet

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Volume

Hint

The formula sheet shows how to calculate the volume of a sphere.

Answer

\(7400\text{ cm}^3\)

Solution

Hint

What is the height of the cylinder itself, without the hemisphere on top?

Answer

\(3900\text{ cm}^3\)

Solution

Hint

Substitute the values you know into the formula provided on the formula sheet.

Answer

\(10.0\text{ cm}\)

Solution

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Equations and Inequations

Hint

Be very careful if you find yourself dividing by a negative number. It is usually best to avoid it.

Answer

\(3>x\) or \(x<3\)

Solution

Hint

The bracket in the first line is being multiplied by negative 2.

Answer

\(x<\frac{3}{2}\) or \(\frac{3}{2}>x\)

Solution

Hint

Multiplying through carefully by the denominator will deal with the fraction.

Answer

\(x=\frac{13}{2}\)

Solution

Hint

Multiplying by the denominator is a good strategy even when the denominator contains a letter.

Answer

\(x=3\)

Solution

Hint

Find an expression for the perimeter of each shape first. You are told they have equal perimeter...

Answer

\(10\text{ cm}\)

Solution

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Circle Geometry

Hint

Look for right-angles, and isosceles triangles.

Answer

\(\angle ABD=61^{\circ}\)

Solution

Hint

Look for right-angles, and isosceles triangles.

Answer

\(\angle PQR = 34^{\circ}\)

Solution

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Scientific Notation

Hint

Make sure you can enter numbers in scientific notation into your calculator to perform calculations.

Answer

\(8.72\times10^{10}\text{ miles}\)

Solution

Hint

Make sure you can enter numbers in scientific notation into your calculator, and consider whether they should be put them in brackets when performing calculators.

Answer

\(18\)

Solution

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Expanding Brackets

Hint

Take care with negatives when expanding brackets.

Answer

\(6x^2+5x-3\)

Solution

Hint

Check the six terms obtained from expanding these brackets are correct carefully before you collect like terms.

Answer

\(x^3+2x^2-13x+10\)

Solution

Hint

Take care with multiplying a negative by a negative.

Answer

\(2x^3-5x^2-13x-4\)

Solution

Hint

Tackle the expansion of \((x-3)(x+1)\) then consider the "\(-\)" that lies in front of it.

Answer

\(-x+3\)

Solution

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Similar Shapes

Hint

Draw the two triangles separately.

Answer

\(x=6\)

Solution

Hint

Obtain the LSF first, and then think about which scale factor is needed.

Answer

\(7500\text{ cm}^3\)

Solution

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Percentages

Hint

"100% + 2.5%"

Answer

\(£344.61\)

Solution

Hint

What does appreciate mean?

Answer

\(£148 000\)

Solution

Hint

What percentage of their original pay is £11.18?

Answer

\(£10.75\)

Solution

Hint

Without a calculator, finding 1% will be awkward. What other percentage could be reached, which would let you then find 100% quickly?

Answer

\(32\)

Solution

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Numerical Fractions

Hint

When multiplying (or dividing) factions, the denominators don't need to be the same. However, you don't want either to be a mixed fraction.

Answer

\(\dfrac{26}{11}\) or \(2\dfrac{4}{11}\)

Solution

Hint

How can a division by a fraction be rewritten as a multiplication?

Answer

\(\dfrac{20}{9}\) or \(2\dfrac{2}{9}\)

Solution

Hint

When adding (or subtracting) fractions, the denominators need to be equal.

Answer

\(\dfrac{193}{30}\) or \(6\dfrac{13}{30}\)

Solution

Hint

"\(\dfrac{3}{4}\) of..." is the same as "\(\dfrac{3}{4}\times\)..."

Answer

\(\dfrac{15}{8}\) or \(1\dfrac{7}{8}\)

Solution

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Pythagoras

Hint

Isolate a right-angled triangle within the diagram to focus on.

Answer

\(6.56\text{ mm}\)

Solution

Hint

Start by calculating the length of line AC, which lies beneath line AD.

Answer

\(8.37\text{ m}\)

Solution

Hint

Since line JF lies East-West, the question is asking whether line FS makes a right-angle with JF.

Answer

No, the Second Buoy is not directly South of the First Buoy (with justification).

Solution

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Straight Lines

Hint

Start by using a formula to find the gradient between the points.

Answer

\(y=2x+8\)

Solution

Hint

Start by using a formula to find the gradient between the points.

Answer

\(y=-\frac{3}{2}x+11\) or \(2y=-3x+22\)

Solution

Hint

First rearrange the equation to make \(y\) the subject: \(y=\dots\)

Answer

\((0,2)\)

Solution

Hint

First rearrange the equation to make \(y\) the subject: \(y=\dots\)

Answer

\(m=\frac{3}{2}\)

Solution

Hint

Start by finding two data points which lie directly on the line of best fit, and write down their coordinates.

Answer

Equation: \(W=10H+10\)

Weight: \(250\text{ grams}\)

Solution

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Functions

Hint

Use brackets when substituting values into functions.

Answer

\(68\)

Solution

Hint

Take care with negatives.

Answer

\(-2\)

Solution

Hint

The questions is not asking for 17 to be substituted into the function in place of \(x\).

Answer

\(a=7\)

Solution

Hint

You can always check your answer for \(t\) is correct by substituting into into the function afterwards.

Answer

\(t=-\frac{4}{3}\)

Solution

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Factorising

Hint

There are three kinds of factorising to consider in turn: common factor, difference of two squares and trinomials/double brackets. Sometimes more than one kind is needed in a single factorising question.

Answer

\(2(x+3)(x-3)\)

Solution

Hint

\(9m^2\) can be written as \((\dots)^2\)?

Answer

\((3m+5)(3m-5)\)

Solution

Hint

"Think of two numbers that multiple to give... and add to give..."

Answer

\((x-4)(x-2)\)

Solution

Hint

It can help to write out all of the possible pairs of whole numbers that multiply to give -20.

Answer

\((y+5)(y-4)\)

Solution

Hint

Always look to take a common factor first. However, there may still be more to do afterwards...

Answer

\(3(x+3)(x-1)\)

Solution

Hint

One approach is to find a pair of numbers that multiply to give 5, but that add to give -11 when one of them is multiplied by 2 first.

Answer

\((2x-1)(x-5)\)

Solution

Hint

Once you think you have the correct answer, you can always multiply your brackets out again to check you get back to the original expression.

Answer

\((3x-1)(x+2)\)

Solution

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Statistics

Hint

Start by putting the numbers in order...

Answer

1. Median = 21, interquartile range = 16.
2. Compare your comments to those in the solution, or ask your teacher to check them.

Solution

Hint

There are two formulae on the formula sheet for standard deviation. Check your notes to see which one your teacher taught you how to use.

Answer

1. Mean = 17, standard deviation = 9.7.
2. Compare your comments to those in the solution, or ask your teacher to check them.

Solution

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Quadratics

Hint

To solve a quadratic equation, you will usually want to factorise first.

Answer

\(x=-2,x=5\)

Solution

Hint

"\(y\)-intercepts when \(x=0\)..." "Roots when \(y=0\)..." "The \(x\)-coordinate of the turning point lies in the middle of the roots... "

Answer

\(y\)-intercept at \((0,-15)\) Roots at \((5,0)\) and \((-3,0)\) Minimum turning point at \((1,-16)\)

Solution

Hint

Make sure you label all your coordinates on the sketch. If any numbers don't make sense, check to see if you have done something wrong.

Answer

\(y\)-intercept at \((0,-7)\) Roots at \((1,0)\) and \((-7,0)\) Minimum turning point at \((-3,-16)\)

Solution

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Arcs and Sectors

Hint

An arc is a fraction of a circumference.

Answer

\(59.9\text{ cm}\)

Solution

Hint

This is another example of substituting into a formula then rearranging.

Answer

\(60.8^{\circ}\)

Solution

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Changing the Subject

Hint

Deal with \(r\) being within a fraction first.

Answer

\(r=\dfrac{3P+k}{2}\)

Solution

Hint

Deal with \(m\) being within a square root first.

Answer

\(m=t^2+a^2\)

Solution

Hint

As with the previous example, the letter is within a square root. But that doesn't mean you should deal with the square root right away.

Answer

\(k=(y-pr)^2\)

Solution

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Surds

Hint

Think of the highest square factor of each number within a root.

Answer

\(11\sqrt{3}\)

Solution

Hint

Take care not to forget the 3 in front of the first surd.

Answer

\(19\sqrt{2}\)

Solution

Hint

Multiply by \(\dfrac{?}{?}\)...

Answer

\(\dfrac{3\sqrt{35}}{7}\)

Solution

Hint

Always check whether the surd can be simplified.

Answer

\(\dfrac{2\sqrt{3}}{3}\)

Solution

Hint

Numerical terms can be collected together, and surds which are alike can be collected together. It isn't always possible to collect things together...

Answer

\(Answer\)

Solution

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Indices

Hint

Learn your index laws carefully.

Answer

\(6x^2\)

Solution

Hint

Remember that a negative power means division.

Answer

\(\dfrac{4}{x^2}\)

Solution

Hint

The order of operations means apply the power (of a power) before the multiplication (of powers).

Answer

\(\dfrac{1}{p^5}\)

Solution

Hint

Remember that both the \(y\) term and the fraction in front are being squared.

Answer

\(\dfrac{9}{16}y^{10}\)

Solution

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Trig Graphs

Hint

One of the letters is the amplitude, the other is the number of waves in 360 degrees.

Answer

\(a=5,b=3\)

Solution

Hint

The graph is only shown up to 180 degrees...

Answer

\(a=3,b=2\)

Solution

Hint

The graphs has been moved up, compared to a \(y=a\sin{x^\circ}\) graph.

Answer

\(a=3,b=2\)

Solution

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Simultaneous Equations

Hint

The equations must be scaled first.

Answer

\(x=5,y=1\)

Solution

Hint

Take care with negatives, as usual.

Answer

\(p=3,q=-2\)

Solution

Hint

Communicate your answer fully at the end.

Answer

A full price ticket costs £7. A concession ticket costs £5.

Solution

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Area and Sine Rules

Hint

Make your you give a complete answer, including units.

Answer

\(11.6\text{ m}^2\)

Solution

Hint

Remember that the Sine Rule can be written in two different ways. Do you want sides or angles on top?

Answer

\(3.7 \text{ cm}\)

Solution

Hint

Remember that the Sine Rule can be written in two different ways. Do you want sides or angles on top?

Answer

\(34.0^\circ\)

Solution

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Algebraic Fractions

Hint

In an SQA exam question, it is likely that the numerator and denominator will have something in common after factorising.

Answer

\(\dfrac{x-5}{x-3}\)

Solution

Hint

To simplify an algebraic fraction like this, you should first factorise.

Answer

\(\dfrac{x-5}{3x}\)

Solution

Hint

To simplify an algebraic fraction like this, you should first factorise.

Answer

\(\dfrac{x+10}{x-2}\)

Solution

NOTE: To use the widgets in the compiled HTML file, you need to have a JavaScript-enabled browser.

Fill-In-The-Blanks (fitb())

Create fill-in-the-blank questions using fitb(), providing the answer as the first argument.

• 2 + 2 is

You can also create these questions dynamically, using variables from your R session.

• The square root of 64 is:

The blanks are case-sensitive; if you don't care about case, use the argument ignore_case = TRUE.

• What is the letter after D?

If you want to ignore differences in whitespace use, use the argument ignore_ws = TRUE (which is the default) and include spaces in your answer anywhere they could be acceptable.

• How do you load the tidyverse package?

You can set more than one possible correct answer by setting the answers as a vector.

• Type a vowel:

You can use regular expressions to test answers against more complex rules.

• Type any 3 letters:

Multiple Choice (mcq())

• "Never gonna give you up, never gonna: let you goturn you downrun awaylet you down"
• "I bless the rainsguess it rainssense the rain down in Africa" -Toto

True or False (torf())

• True or False? You can permute values in a vector using sample(). TRUEFALSE

Longer MCQs (longmcq())

When your answers are very long, sometimes a drop-down select box gets formatted oddly. You can use longmcq() to deal with this. Since the answers are long, It's probably best to set up the options inside an R chunk with echo=FALSE.

What is a p-value?

the probability that the null hypothesis is true the probability of the observed, or more extreme, data, under the assumption that the null-hypothesis is true the probability of making an error in your conclusion

What is true about a 95% confidence interval of the mean?

if you repeated the process many times, 95% of intervals calculated in this way contain the true mean there is a 95% probability that the true mean lies within this range 95% of the data fall within this range

Checked sections

Create sections with the class webex-check to add a button that hides feedback until it is pressed. Add the class webex-box to draw a box around the section (or use your own styles).

I am going to learn a lot: TRUEFALSE

What is a p-value?

the probability that the null hypothesis is true the probability of the observed, or more extreme, data, under the assumption that the null-hypothesis is true the probability of making an error in your conclusion

Hidden solutions and hints

You can fence off a solution area that will be hidden behind a button using hide() before the solution and unhide() after, each as inline R code. Pass the text you want to appear on the button to the hide() function.

If the solution is an RMarkdown code chunk, instead of using hide() and unhide(), simply set the webex.hide chunk option to TRUE, or set it to the string you wish to display on the button.

Recreate the scatterplot below, using the built-in cars dataset.

I need a hint

See the documentation for plot() (?plot)

Click here to see the solution

plot(cars$speed, cars$dist)